Group #13093004 — subgroups

Subgroups — #13093004

Order 1472.

250 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 95 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#121	C₂	1	1	yes	maximal, characteristic, complemented
2	#10625110	C₂	1	2	yes	maximal, complemented
3	#10625108	C₂	1	2	yes	maximal, complemented
4	D₄×Dic₂₃	C₂	1	2	yes	maximal, complemented
5	C₂×(C₂×D₂₃)⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
6	#10625095	C₂	1	1	yes	maximal, characteristic, complemented
7	#10625069	C₂	1	2	yes	maximal, complemented
8	#10625057	C₂	1	2	yes	maximal, complemented
9	#10625056	C₂	1	2	yes	maximal, complemented
10	C₂³×D₂₃	C₂²	1	1	yes	characteristic
11	C₂×D₄×C₂₃	C₂²	1	2	yes	—
12	C₂×D₄×C₂₃	C₂²	1	1	yes	characteristic, complemented
13	C₂²×C₄×C₂₃	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊D₂₃	C₂²	1	2	yes	—
15	C₂×C₂²⋊D₂₃	C₂²	1	2	yes	—
16	C₂²×Dic₂₃	C₂²	1	1	yes	characteristic
17	C₂²×Dic₂₃	C₂²	1	2	yes	complemented
18	C₂×Dic₄₆	C₂²	1	1	yes	characteristic, complemented
19	C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic, complemented
20	C₂²⋊C₄×C₂₃	C₂²	1	2	yes	complemented
21	C₂²⋊Dic₂₃	C₂²	1	2	yes	—
22	C₂²⋊Dic₂₃	C₂²	1	2	yes	complemented
23	(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
24	(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
25	(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
26	(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
27	C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic, complemented
28	C₄⋊Dic₂₃	C₂²	1	2	yes	complemented
29	Dic₂₃⋊C₄	C₂²	1	2	yes	complemented
30	C₄×Dic₂₃	C₂²	1	2	yes	complemented
31	C₂³×C₂₃	C₂³	1	2	yes	—
32	C₂³×C₂₃	C₂³	1	1	yes	characteristic
33	C₂²×D₂₃	—	2	2	no	—
34	C₂²×D₂₃	—	4	1	no	—
35	C₂²×D₂₃	—	4	1	no	—
36	C₂²×D₂₃	C₂³	1	2	yes	—
37	D₄×C₂₃	—	2	4	no	—
38	D₄×C₂₃	—	4	1	no	—
39	C₂×C₄×C₂₃	—	4	1	no	—
40	C₂×C₄×C₂₃	C₂³	1	2	yes	—
41	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
42	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
43	C₂²⋊D₂₃	—	4	2	no	—
44	C₂²⋊D₂₃	—	4	2	no	—
45	C₂×Dic₂₃	—	4	1	no	—
46	C₂×Dic₂₃	—	4	2	no	—
47	C₂×Dic₂₃	C₂³	1	2	yes	—
48	C₂×Dic₂₃	C₂³	1	2	yes	—
49	C₂×Dic₂₃	C₂³	1	2	yes	—
50	Dic₄₆	—	4	1	no	—
51	C₂²×C₂₃	—	2	1	no	—
52	C₂²×C₂₃	—	2	1	no	—
53	C₂²×C₂₃	—	2	2	no	—
54	C₂²×C₂₃	—	2	2	no	—
55	C₂²×C₂₃	—	2	2	no	—
56	C₂²×C₂₃	#41	1	2	yes	—
57	C₂²×C₂₃	C₂⁴	1	1	yes	characteristic
58	C₂×D₂₃	—	2	2	no	—
59	C₂×D₂₃	—	2	2	no	—
60	C₂×D₂₃	—	2	2	no	—
61	C₂×D₂₃	—	4	4	no	—
62	C₄×C₂₃	—	2	2	no	—
63	C₄×C₂₃	—	2	1	no	—
64	C₄×C₂₃	—	2	1	no	—
65	Dic₂₃	—	2	2	no	—
66	Dic₂₃	—	2	2	no	—
67	Dic₂₃	—	2	2	no	—
68	#538	—	23	1	no	maximal, Sylow
69	C₂×C₂₃	—	2	2	no	—
70	C₂×C₂₃	—	4	2	no	—
71	C₂×C₂₃	C₂×#41	1	1	yes	characteristic
72	C₂×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
73	C₂×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
74	D₂₃	—	4	2	no	—
75	#126	—	23	2	no	—
76	#124	—	23	2	no	—
77	#123	—	23	2	no	—
78	C₂²⋊Q₈	—	23	1	no	—
79	#121	—	23	2	no	—
80	#121	D₂₃	1	1	yes	characteristic, complemented
81	#120	—	23	2	no	—
82	C₄×D₄	—	23	2	no	—
83	C₂×C₂²⋊C₄	—	23	1	no	—
84	C₂₃	#538	1	1	yes	characteristic, complemented, Sylow
85	C₂⁴	—	23	1	no	—
86	C₂×Q₈	—	23	1	no	—
87	C₂×D₄	—	23	2	no	—
88	C₂×D₄	—	23	2	no	—
89	C₂×D₄	C₂×D₂₃	1	2	yes	—
90	C₂×D₄	C₂×D₂₃	1	1	yes	characteristic, complemented
91	C₂²×C₄	—	23	1	no	—
92	C₂²×C₄	—	23	2	no	—
93	C₂²×C₄	C₂×D₂₃	1	1	yes	characteristic
94	C₄⋊C₄	—	23	2	no	—
95	C₄⋊C₄	—	23	2	no	—
96	C₄⋊C₄	—	23	1	no	—
97	C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic, complemented
98	C₂²⋊C₄	—	23	2	no	—
99	C₂²⋊C₄	—	23	2	no	—
100	C₂²⋊C₄	—	23	2	no	—
101	C₂²⋊C₄	—	23	2	no	—
102	C₂²⋊C₄	—	23	2	no	—
103	C₂²⋊C₄	—	23	2	no	—
104	C₂²⋊C₄	C₂×D₂₃	1	2	yes	complemented
105	C₄²	—	23	2	no	—
106	C₂³	—	23	2	no	—
107	C₂³	—	46	2	no	—
108	C₂³	—	92	1	no	—
109	C₂³	—	92	1	no	—
110	C₂³	C₂²×D₂₃	1	2	yes	—
111	C₂³	C₂²×D₂₃	1	1	yes	characteristic
112	Q₈	—	92	1	no	—
113	D₄	—	2	4	no	—
114	D₄	—	4	1	no	—
115	D₄	—	92	2	no	—
116	D₄	—	92	2	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	23	2	no	—
119	C₂×C₄	—	23	2	no	—
120	C₂×C₄	—	23	2	no	—
121	C₂×C₄	—	92	1	no	—
122	C₂×C₄	—	92	2	no	—
123	C₂×C₄	C₂²×D₂₃	1	2	yes	—
124	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
125	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
126	C₂²	—	2	1	no	—
127	C₂²	—	2	1	no	—
128	C₂²	—	2	2	no	—
129	C₂²	—	2	2	no	—
130	C₂²	—	2	2	no	—
131	C₂²	—	46	2	no	—
132	C₂²	—	46	2	no	—
133	C₂²	—	46	2	no	—
134	C₂²	—	92	4	no	—
135	C₂²	#67290	1	2	yes	—
136	C₂²	C₂³×D₂₃	1	1	yes	characteristic
137	C₄	—	2	2	no	—
138	C₄	—	2	1	no	—
139	C₄	—	2	1	no	—
140	C₄	—	46	2	no	—
141	C₄	—	46	2	no	—
142	C₄	—	46	2	no	—
143	C₂	—	2	2	no	—
144	C₂	—	4	2	no	—
145	C₂	—	92	2	no	—
146	C₂	C₂×#67290	1	1	yes	characteristic
147	C₂	#10625155	1	1	yes	characteristic
148	C₂	#10625160	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×#121

C₂

yes

maximal, characteristic, complemented

#10625110

C₂

yes

maximal, complemented

#10625108

C₂

yes

maximal, complemented

D₄×Dic₂₃

C₂

yes

maximal, complemented

C₂×(C₂×D₂₃)⋊C₄

C₂

yes

maximal, characteristic, complemented

#10625095

C₂

yes

maximal, characteristic, complemented

#10625069

C₂

yes

maximal, complemented

#10625057

C₂

yes

maximal, complemented

#10625056

C₂

yes

maximal, complemented

C₂³×D₂₃

C₂²

yes

characteristic

C₂×D₄×C₂₃

C₂²

yes

—

C₂×D₄×C₂₃

C₂²

yes

characteristic, complemented

C₂²×C₄×C₂₃

C₂²

yes

characteristic

C₂×C₂²⋊D₂₃

C₂²

yes

—

C₂×C₂²⋊D₂₃

C₂²

yes

—

C₂²×Dic₂₃

C₂²

yes

characteristic

C₂²×Dic₂₃

C₂²

yes

complemented

C₂×Dic₄₆

C₂²

yes

characteristic, complemented

C₄⋊C₄×C₂₃

C₂²

yes

characteristic, complemented

C₂²⋊C₄×C₂₃

C₂²

yes

complemented

C₂²⋊Dic₂₃

C₂²

yes

—

C₂²⋊Dic₂₃

C₂²

yes

complemented

(C₂×D₂₃)⋊C₄

C₂²

yes

—

(C₂×D₂₃)⋊C₄

C₂²

yes

—

(C₂×D₂₃)⋊C₄

C₂²

yes

—

(C₂×D₂₃)⋊C₄

C₂²

yes

—

C₄⋊Dic₂₃

C₂²

yes

characteristic, complemented

C₄⋊Dic₂₃

C₂²

yes

complemented

Dic₂₃⋊C₄

C₂²

yes

complemented

C₄×Dic₂₃

C₂²

yes

complemented

C₂³×C₂₃

C₂³

yes

—

C₂³×C₂₃

C₂³

yes

characteristic

C₂²×D₂₃

—

C₂²×D₂₃

—

C₂²×D₂₃

—

C₂²×D₂₃

C₂³

yes

—

D₄×C₂₃

—

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂³

yes

—

C₂×C₄×C₂₃

C₂³

yes

characteristic

C₂×C₄×C₂₃

C₂³

yes

characteristic

C₂²⋊D₂₃

—

C₂²⋊D₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

C₂³

yes

—

C₂×Dic₂₃

C₂³

yes

—

C₂×Dic₂₃

C₂³

yes

—

Dic₄₆

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#41

yes

—

C₂²×C₂₃

C₂⁴

yes

characteristic

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

Dic₂₃

—

Dic₂₃

—

Dic₂₃

—

#538

—

maximal, Sylow

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

C₂×#41

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

D₂₃

—

#126

—

#124

—

#123

—

C₂²⋊Q₈

—

#121

—

#121

D₂₃

yes

characteristic, complemented

#120

—

C₄×D₄

—

C₂×C₂²⋊C₄

—

C₂₃

#538

yes

characteristic, complemented, Sylow

C₂⁴

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₂₃

yes

—

C₂×D₄

C₂×D₂₃

yes

characteristic, complemented

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₂₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₂₃

yes

characteristic, complemented

C₂²⋊C₄

—

C₂²⋊C₄

—

100

C₂²⋊C₄

—

101

C₂²⋊C₄

—

102

C₂²⋊C₄

—

103

C₂²⋊C₄

—

104

C₂²⋊C₄

C₂×D₂₃

yes

complemented

105

C₄²

—

106

C₂³

—

107

C₂³

—

108

C₂³

—

109

C₂³

—

110

C₂³

C₂²×D₂₃

yes

—

111

C₂³

C₂²×D₂₃

yes

characteristic

112

Q₈

—

113

D₄

—

114

D₄

—

115

D₄

—

116

D₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

C₂²×D₂₃

yes

—

124

C₂×C₄

C₂²×D₂₃

yes

characteristic

125

C₂×C₄

C₂²×D₂₃

yes

characteristic

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

#67290

yes

—

136

C₂²

C₂³×D₂₃

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₂

—

144

C₂

—

145

C₂

—

146

C₂

C₂×#67290

yes

characteristic

147

C₂

#10625155

yes

characteristic

148

C₂

#10625160

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₃
commutator, socle, cosocle	C₂²×C₂₃	C₂⁴
Fitting	C₂₃×#121	C₂
hypercenter	#121	D₂₃
hypocenter	C₂₃	#538
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ #121

subgroup

G/H

center, Frattini

C₂²

C₂³×D₂₃

commutator, socle, cosocle

C₂²×C₂₃

C₂⁴

Fitting

C₂₃×#121

C₂

D₂₃

C₂₃

G ▷ C₂²×C₂₃ ▷ C₂₃

C₁ ◁ C₂² ◁ #121

N	H ≅ G/N	type
1. C₂₃×#121	C₂	semidirect
2. #10625110	C₂	semidirect
3. #10625108	C₂	semidirect
4. D₄×Dic₂₃	C₂	semidirect
5. C₂×(C₂×D₂₃)⋊C₄	C₂	semidirect
6. #10625095	C₂	semidirect
7. #10625069	C₂	semidirect
8. #10625057	C₂	semidirect
9. #10625056	C₂	semidirect
12. C₂×D₄×C₂₃	C₂²	semidirect
17. C₂²×Dic₂₃	C₂²	semidirect
18. C₂×Dic₄₆	C₂²	semidirect
19. C₄⋊C₄×C₂₃	C₂²	semidirect
20. C₂²⋊C₄×C₂₃	C₂²	semidirect
22. C₂²⋊Dic₂₃	C₂²	semidirect
27. C₄⋊Dic₂₃	C₂²	semidirect
28. C₄⋊Dic₂₃	C₂²	semidirect
29. Dic₂₃⋊C₄	C₂²	semidirect
30. C₄×Dic₂₃	C₂²	semidirect
80. #121	D₂₃	semidirect
84. C₂₃	#538	semidirect
90. C₂×D₄	C₂×D₂₃	semidirect
97. C₄⋊C₄	C₂×D₂₃	semidirect
104. C₂²⋊C₄	C₂×D₂₃	semidirect

H ≅ G/N

type

1. C₂₃×#121

C₂

semidirect

2. #10625110

C₂

semidirect

3. #10625108

C₂

semidirect

4. D₄×Dic₂₃

C₂

semidirect

5. C₂×(C₂×D₂₃)⋊C₄

C₂

semidirect

6. #10625095

C₂

semidirect

7. #10625069

C₂

semidirect

8. #10625057

C₂

semidirect

9. #10625056

C₂

semidirect

12. C₂×D₄×C₂₃

C₂²

semidirect

17. C₂²×Dic₂₃

C₂²

semidirect

18. C₂×Dic₄₆

C₂²

semidirect

19. C₄⋊C₄×C₂₃

C₂²

semidirect

20. C₂²⋊C₄×C₂₃

C₂²

semidirect

22. C₂²⋊Dic₂₃

C₂²

semidirect

27. C₄⋊Dic₂₃

C₂²

semidirect

28. C₄⋊Dic₂₃

C₂²

semidirect

29. Dic₂₃⋊C₄

C₂²

semidirect

30. C₄×Dic₂₃

C₂²

semidirect

80. #121

D₂₃

semidirect

84. C₂₃

#538

semidirect

90. C₂×D₄

C₂×D₂₃

semidirect

97. C₄⋊C₄

C₂×D₂₃

semidirect

104. C₂²⋊C₄

C₂×D₂₃

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂₃×#121	2
#10625110	2
#10625108	2
D₄×Dic₂₃	2
C₂×(C₂×D₂₃)⋊C₄	2
#10625095	2
#10625069	2
#10625057	2
#10625056	2
#538	23

subgroup

index

C₂₃×#121

#10625110

#10625108

D₄×Dic₂₃

C₂×(C₂×D₂₃)⋊C₄

Subgroups — #13093004

Conjugacy classes of subgroups

Complements

Maximal subgroups